Massive Black Hole Binary EvolutionCoalescence of binary supermassive black holes (SBHs) would constitute the strongest sources of gravitational waves to be observed by LISA. While the formation of binary SBHs during galaxy mergers is almost inevitable, coalescence requires that the separation between binary components first drop by a few orders of magnitude, due presumably to interaction of the binary with stars and gas in a galactic nucleus. This article reviews the observational evidence for binary SBHs and discusses how they would evolve. No completely convincing case of a bound, binary SBH has yet been found, although a handful of systems (e.g. interacting galaxies; remnants of galaxy mergers) are now believed to contain two SBHs at projected separations of &lt;~ 1kpc. N-body studies of binary evolution in gas-free galaxies have reached large enough particle numbers to reproduce the slow, “diffusive” refilling of the binary’s loss cone that is believed to characterize binary evolution in real galactic nuclei. While some of the results of these simulations - e.g. the binary hardening rate and eccentricity evolution - are strongly N-dependent, others - e.g. the “damage” inflicted by the binary on the nucleus - are not. Luminous early-type galaxies often exhibit depleted cores with masses of ~ 1-2 times the mass of their nuclear SBHs, consistent with the predictions of the binary model. Studies of the interaction of massive binaries with gas are still in their infancy, although much progress is expected in the near future. Binary coalescence has a large influence on the spins of SBHs, even for mass ratios as extreme as 10:1, and evidence of spin-flips may have been observed.2005-10-292005-11-22lrr-2005-8Max Planck Institute for Gravitational Physics